PaperlayerView exact revision on arXiv

Source-linked AI summary

The structure and stability of persistence modules

Frederic Chazal, Vin de Silva, Marc Glisse, Steve Oudot

arXiv:1207.3674v3math.ATcs.CGmath.CT

TL;DR

Persistence modules arising in applications are often finite, but idealized models may not satisfy finiteness conditions. This paper develops measure-based persistence diagrams and related algebraic constructions, showing that the theory still applies to some infinite-dimensional modules and supports stability results.

  • Problem

    Applied persistence modules are often finite, but idealized continuous models make finiteness unnatural and difficult to enforce while retaining the main theoretical results remains desirable.

  • Method

    The paper defines persistence diagrams from interval decompositions and develops r-measures, measure–diagram equivalences, and simplified quiver-representation calculations.

  • Results

    The framework proves stability inequalities for persistence-module measures and applies even when individual vector spaces are infinite-dimensional.

  • Takeaways & Limitations

    Weaker tameness conditions can support persistence-diagram theory beyond the finite-dimensional setting encountered in standard applied examples.

  • Takeaways & Limitations

    The treatment of locally compact polyhedra with proper functions is left as an exercise, including locating possible singularities of the measures.

Abstract

from arXiv · show

We give a self-contained treatment of the theory of persistence modules indexed over the real line. We give new proofs of the standard results. Persistence diagrams are constructed using measure theory. Linear algebra lemmas are simplified using a new notation for calculations on quiver representations. We show that the stringent finiteness conditions required by traditional methods are not necessary to prove the existence and stability of the persistence diagram. We introduce weaker hypotheses for taming persistence modules, which are met in practice and are strong enough for the theory still to work. The constructions and proofs enabled by our framework are, we claim, cleaner and simpler.

1. Persistence Modules

Persistence modules organize vector spaces and compatible linear maps over ordered index sets, with the real line as the main setting. The section develops interval-based descriptions where available, while extending persistence diagrams and stability beyond finite data through q-tameness.

  • Persistence modules: A persistence module is a functor from an ordered index set to vector spaces, assigning spaces and compatible maps between indices.The real line is treated as a category with one morphism whenever s ≤ t, so composition and identity laws are built into the structure.
  • Persistence modules: Sublevelset filtrations produce persistence modules by applying homology or another functor to nested spaces X_t.For a function f, X_t consists of points with f(x) ≤ t, and inclusions induce the maps between homology groups.
  • Applications and finiteness: Finite simplicial-complex filtrations reduce persistence to finite-dimensional linear diagrams, yielding compact, computable, and 1-Lipschitz persistence descriptions.The reduction occurs at finitely many critical values where the complex changes by adding cells.
  • Tameness: The paper motivates q-tame modules because idealized theoretical models need not be finite, yet persistence diagrams should retain existence and stability properties.q-tame modules are defined by finite-rank persistence maps for every s < t and occur, for example, in sublevelset homology of continuous functions on finite simplicial complexes.
  • Interval modules: Interval modules are the basic building blocks of persistence, but some modules do not decompose into intervals.When an interval decomposition exists, its interval multiset is an isomorphism invariant; the paper also develops rectangle measures for the non-decomposable case.
  • Interval modules: Decorated endpoints distinguish open, closed, and half-open intervals, while persistence-module categories support homomorphisms, kernels, images, cokernels, and zero objects.Decorations also support the point representation of intervals and the measure-theoretic treatment of persistence diagrams.

2. Rectangle Measures

The paper defines persistence diagrams by assigning integer-valued rectangle measures to persistence modules, then recovering decorated points from rectangle counts. For decomposable modules, this measure counts interval summands and agrees with the existing diagram definition.

  • Constructing the persistence measure: Persistence modules define integer-valued measures on rectangles, allowing diagram reconstruction even when decomposability is unknown.The measure is additive under rectangle tilings, with decorated points resolving boundary-membership ambiguities.
  • Interval modules: For an interval module J, the rectangle measure is 1 exactly when [b,c] ⊆ J ⊆ (a,d), and otherwise it is 0.Geometrically, interior points are detected regardless of decoration, while boundary points require inward-facing ticks.
  • Decomposable modules: For decomposable persistence modules, the rectangle measure equals the number of interval summands whose decorated points lie in the rectangle.Thus the measure-based definition agrees with the traditional persistence diagram in the decomposable case.
  • Diagram construction: The strategy is to construct µV and define a multiset Dgm(V) whose decorated points reproduce every rectangle measure.Existence and uniqueness of this multiset are supplied by the finite, additive measure representation theorem.
  • Additivity: The persistence measure is additive under both horizontal and vertical rectangle splits.The paper gives three proofs: a general quiver-calculation proof, a finite-dimensional proof using alternating sums, and a decomposable-module proof by counting decorated points.

2.3. Abstract r-measures.

Abstract r-measures formalize rectangle-counting functions independently of persistence modules. Their finite additivity and monotonicity support a bijection between finite r-measures and locally finite decorated multisets.

  • Definition: An r-measure maps rectangles in D to nonnegative integers or infinity and is additive under horizontal and vertical splitting.The domain consists of closed rectangles contained in D.
  • Basic properties: Every r-measure is finitely additive for arbitrary finite decompositions and monotone under inclusion.Subadditivity follows when one rectangle is contained in a finite union of rectangles.
  • Equivalence theorem: Finite r-measures correspond bijectively to locally finite multisets of decorated points.Theorem 2.8 establishes both existence and uniqueness of the multiset representation.
  • Diagram representation: The resulting diagram recovers the measure of every rectangle by counting its decorated points.The construction is first checked for additivity because each decorated point belongs to exactly one piece after a split.
  • Construction and proof: The multiset is constructed through a multiplicity function defined as the minimum measure of rectangles containing each decorated point.The proof then verifies the reconstruction formula by induction on rectangle measure, using repeated quadrant subdivision in the non-terminating case.

2.5. Non-finite measures.

The framework extends diagram construction to r-measures that are not finite everywhere by restricting attention to their finite r-interior. The resulting diagrams retain exactly the finite-measure information and extend naturally to infinite rectangles and points at infinity.

  • General diagrams: A uniquely defined locally finite decorated diagram exists on the finite r-interior, and every rectangle whose decorated points lie there has finite measure.This extends the finite-measure correspondence without requiring global finiteness.
  • Information content: The measure contains no information beyond the diagram: finite rectangles are recovered by counting decorated points, while all other rectangles have infinite measure.The paper explicitly characterizes this as the absence of hidden information.
  • Extended plane: The construction extends to the extended plane by applying the finite theorem locally and gluing the compatible diagrams on overlapping finite-measure rectangles.Infinite rectangles are handled by setting V_-∞ and V_+∞ to zero in the persistence-module formulas.
  • Finite regions: For a general r-measure, the finite r-interior contains precisely the decorated points lying in some finite-measure rectangle.The undecorated finite interior is obtained by restricting to rectangle interiors and forgetting decorations.
  • Persistence modules: For decomposable modules over the extended line, µV(R) counts interval summands corresponding to decorated points in R.The interval-point correspondence and the earlier counting result extend to infinite endpoints.

2.7. Diagrams of persistence modules.

The section develops persistence diagrams through persistence measures, allowing diagrams for non-decomposable modules and introducing tameness conditions that guarantee existence on specified regions.

  • Diagram construction: Persistence diagrams can be defined indirectly through persistence measures, extending the construction beyond modules decomposable into intervals.The measure-based method defines diagrams where the measure is finite.
  • Diagram construction: For decomposable modules, the interval multiset agrees with the measure-derived persistence diagram wherever the latter is defined.This agreement holds on the finite r-interior of the persistence measure.
  • Examples: The Webb example is not interval-decomposable, yet its persistence measure is finite away from the singular point p´8, 0`q, where its diagram is defined.The diagram contains points of the form p´n`, 0`q for n = 1, 2, 3, . . . .
  • Tameness conditions: Four weaker tameness notions control measure finiteness on quadrants, horizontal strips, vertical strips, or finite rectangles, thereby determining where diagrams exist.Each condition includes the finite part of the plane away from the diagonal; they differ at infinity.
  • Tameness conditions: The inclusions are strict: q-tame implies both h-tame and v-tame, while either h-tame or v-tame implies r-tame.The paper leaves open whether h-tame ∩ v-tame equals r-tame.
  • Applications: Persistent homology of a finite polyhedron with a continuous function is q-tame, while a proper continuous function on a locally compact polyhedron yields h-, v-, and r-tameness.These results allow individual homology spaces to be infinite-dimensional in the finite-polyhedron case.

2.9. Finite approximations.

Finite approximations recover the part of a persistence module that can be observed on finite index sets, but the persistence measure may not determine all global structure.

  • Limits of observability: The measure cannot distinguish many nonisomorphic persistence modules when it is infinite on every rectangle.In such cases, the r-measure provides a limited view outside the finite r-interior.
  • Limits of observability: If diagram points approach r from below and above, the measure alone cannot determine the multiplicity of the diagonal point pr´, r`q.This is a specific ambiguity in recovering diagonal multiplicities from limiting behavior.
  • Finite observability: The persistence measure recovers all information obtainable by restricting a module to any finite index set.For finite T, interval multiplicities in the restricted barcode equal measure values on corresponding rectangles.
  • Finite observability: This recovery acts as a snapping principle: decorated diagram points are counted in rectangles immediately below and to the left of grid vertices.Points in the remaining triangular regions do not appear in the discretised diagram.
  • Finitely observable settings: For finite-observable settings, finite critical sets make the module constant between critical values and determine a finite-type decomposition.Compact manifolds with Morse functions and compact polyhedra with piecewise-linear functions are given as examples.

3. Interleaving

Section 3 develops interleavings as approximate isomorphisms for uncertain data and proves that interleaved modules admit continuous interpolations. It also characterizes interleavings through modules over unions of shifted diagonals.

  • Interleavings: δ-interleaving weakens isomorphism by quantifying uncertainty when persistence-module data contain noise.Interleavings arise naturally when input data are known only up to bounded error.
  • Interleavings: A degree-δ homomorphism shifts each index t to t+δ through a commuting family of linear maps.For δ ≥ 0, the persistence shift map is the central degree-δ endomorphism.
  • Interleavings: Two modules are δ-interleaved when compatible maps exist in both directions and their composites equal the corresponding 2δ shifts.The defining diagrams must commute for all eligible parameter values.
  • Interleavings: A pair of modules is |y−x|-interleaved exactly when it extends to a module over the union of shifted diagonals ∆x and ∆y.Each shifted diagonal is canonically identified with the real line, so restrictions recover the original modules.
  • Interpolation: The interpolation lemma constructs a one-parameter family between δ-interleaved modules whose members are |x−y|-interleaved.The extension is constructed via an image or cokernel of module maps; the extension is not unique.
  • Interpolation: The image construction is structurally the simplest of the three interpolations, while kernel and cokernel constructions contain surplus homological information.The three constructions correspond to image, kernel, and cokernel interpolations.

4. The Isometry Theorem

Section 4 relates persistence modules to persistence diagrams through interleaving and bottleneck distances. Its principal results are a measure-theoretic stability theorem and a converse inequality yielding an isometry for q-tame modules.

  • The Isometry Theorem: The stability theorem is formulated as a statement about persistence measures and their diagrams.The proof closely follows the original stability proof while emphasizing measures.
  • The Isometry Theorem: The converse inequality, together with stability, makes q-tame persistence modules isometric to locally finite persistence diagrams.This result extends earlier formulations associated with stronger finiteness conditions.

4.1. The interleaving distance.

Section 4.1 defines interleaving distance as the infimum of admissible interleaving parameters and establishes its basic metric-like properties. It also explains why the distance is a pseudometric rather than a true metric.

  • Definition and properties: The infimum defining interleaving distance need not be attained, motivating the notion of δ+-interleaving.δ+-interleaving requires (δ+ε)-interleavings for every ε>0 and need not imply δ-interleaving.
  • Definition and properties: If no δ-interleaving exists for any δ, the interleaving distance is infinite.A δ-interleaving also implies a (δ+ε)-interleaving for every ε>0.
  • Definition and properties: The interleaving distance satisfies the triangle inequality by composing interleavings.A δ1-interleaving followed by a δ2-interleaving produces a (δ1+δ2)-interleaving.
  • Definition and properties: The distance is a pseudometric because distance zero does not generally imply isomorphism.For q-tame modules, distance zero is equivalent to equality of undecorated persistence diagrams.
  • Definition and properties: The distance between direct sums is bounded by the maximum distance among corresponding summands.The same bound extends to families indexed by a common set through direct-sum interleavings.

4.2. The bottleneck distance.

Section 4.2 defines bottleneck distance using partial matchings in the extended half-plane and relates it to interval-module interleavings. For decomposable modules, a bottleneck matching yields an interleaving bound.

  • Interval modules: For interval modules, interleaving distance is bounded above by the ℓ∞ distance between their interval endpoints.Equality holds when the intervals overlap sufficiently, namely when each closure meets the midpoint of the other.
  • Interval modules: For an interval module, the interleaving distance to zero is controlled by half its interval length, with infinite intervals giving infinite distance.The distance is infinite when the interval is infinite.
  • Definition: Bottleneck distance is the infimum δ for which a δ-matching exists between two multisets of diagram points.Point-to-point distances use the ℓ∞ metric, while points near the diagonal may be matched to it.
  • Metric properties: Finite bottleneck distance requires matching cardinalities to agree separately across the three strata at infinity.Points in different strata have infinite distance and therefore must be matched within their own stratum.
  • Metric properties: The bottleneck distance satisfies the triangle inequality by composing partial matchings.The composite matching has displacement at most δ1+δ2.
  • Comparison theorem: For decomposable persistence modules, every δ-matching between persistence diagrams induces a δ-interleaving.The result combines interval-module bounds with the direct-sum distance inequality.
  • Comparison theorem: The resulting comparison gives dipU, Vq ≤ dbpdgmpUq, dgmpVqq for decomposable modules.Together with the converse inequality, the paper obtains equality in the isometry theorem.

4.3. The bottleneck distance (continued).

The bottleneck distance is defined through δ-matchings, and a compactness argument shows that matchings approaching δ yield an exact δ-matching.

  • The bottleneck distance is the infimum of δ values for which a δ-matching exists between two locally finite multisets.
  • If every η greater than δ admits an η-matching between A and B, then A and B admit a δ-matching.
  • The proof constructs a limiting matching from matchings with parameters δ + 1/n using indicator functions and nested infinite subsets.
  • Nested-subset selection makes each indicator value stabilize, producing a candidate matching whose validity is checked through finite constraints.
  • The limiting indicator satisfies uniqueness and coverage conditions on both multisets, thereby defining a δ-matching.
  • The argument is a direct compactness proof and can also be viewed as an instance of first-order logical compactness.

4.4. The isometry theorem.

For q-tame persistence modules, interleaving and bottleneck distances coincide. The theorem combines the usual stability inequality with its converse, proved by extending the decomposable case.

  • For q-tame modules U and V, the isometry theorem states that interleaving distance equals bottleneck distance between their persistence diagrams.
  • The stability theorem provides the inequality d_I(U,V) ≥ d_B(dgm(U),dgm(V)).
  • The converse stability theorem provides the reverse inequality d_I(U,V) ≤ d_B(dgm(U),dgm(V)).
  • The converse proof extends the result from decomposable modules to q-tame modules without a known interval decomposition.

4.5. The converse stability theorem.

Smoothing approximates q-tame persistence modules by locally finite, decomposable modules while shifting persistence diagrams predictably. This enables the converse stability theorem without assuming interval decompositions initially.

  • The ε-smoothing V_ε is defined as the image of a structure map and shrinks interval supports by ε at both ends.
  • Smoothing translates diagram points by T_ε:(p,q) ↦ (p+ε,q−ε) above the line Δ_ε.
  • Information from Dgm(V) below Δ_ε is lost in Dgm(V_ε).
  • Every q-tame module becomes locally finite and therefore interval-decomposable after ε-smoothing.
  • Applying the decomposable converse theorem to U_ε and V_ε, then letting ε approach zero, yields d_I(U,V) ≤ d_B(dgm(U),dgm(V)).
  • A q-tame module is characterized by being approximable in interleaving distance by locally finite modules.
  • A q-tame module need not admit an interval decomposition, as shown by a cartesian-product example.

4.6. The stability theorem.

The stability theorem derives local measure inequalities from δ-interleavings and converts them into δ-matchings between persistence diagrams. The proof uses interpolation, box inequalities, continuity, and compactness.

  • δ-interleaved q-tame modules U and V have persistence diagrams related by a δ-matching.
  • The proof embeds the modules in a one-parameter family and uses the box lemma to compare their persistence measures locally.
  • The δ-thickening of a rectangle expands its boundary coordinates by δ according to the stated construction.
  • The box lemma bounds each module’s measure on a rectangle by the other module’s measure on its δ-thickening.
  • These local inequalities extend to rectangles reaching positive or negative infinity, together with equality of total measures at infinity.
  • A continuity argument converts the measure comparisons into a diagram matching, while compactness handles diagrams of infinite cardinality.

4.7. The measure stability theorem.

The measure stability theorem uses a one-parameter family of measures satisfying a box inequality to obtain matchings between persistence diagrams, first for finite measures and then without finite-cardinality assumptions. This establishes stability for q-tame persistence modules and supports the isometry theorem.

  • Theorem 4.25: A box inequality for a one-parameter family of finite r-measures yields a δ-matching between the endpoint diagrams.The rectangles must remain in the open domain after |y−x|-thickening.
  • Finite measures: The proof first bounds the Hausdorff distance between diagrams at parameters x and y by |y−x|.Points sufficiently far from the domain boundary have nearby counterparts in the other diagram.
  • Proof strategy: Finite cardinality enables local matching, while a compact exhaustion and limiting partial matchings remove that assumption.Local finiteness makes the diagrams countable and permits the limit construction.
  • Points at infinity: The same three-part argument extends from the standard plane to lines at infinity, while corner multiplicities agree directly.At infinity, squares become intervals; interpolating measures are unnecessary at the four corners.
  • Consequences: The resulting stability theorem for finite measures implies stability for q-tame modules and then the isometry theorem.The proof framework also generalises to measures that are not necessarily finite.

4.8. The measure stability theorem (continued).

The continuation extends measure stability to non-finite measures by restricting them to compatible relatively compact domains, applying the finite theorem, and passing to a limit. It also formalizes matchings across unequal domains.

  • Unequal domains: For unequal domains, a δ-matching requires domain compatibility, bounded pairwise distance, and cross-boundary matching conditions.A point may remain unmatched only when it lies near the other diagram’s domain boundary.
  • Unequal domains: The triangle inequality composes a δ1-matching and a δ2-matching into a (δ1+δ2)-matching.The conclusion applies to multisets in three potentially different open domains.
  • Main theorem: The full theorem gives a δ-matching between the endpoint diagrams on their finite interiors under the global box inequality.The measures are defined on R2, and the result uses the finite interiors F0 and Fδ.
  • Proof conclusion: The construction concludes stability for measures without requiring the measures themselves to be finite.A sequence of δ-matchings is formed using restrictions with radii tending to infinity and trimming parameters tending to zero.
  • Non-finite measures: For non-finite measures, compatible relatively compact subsets are chosen so their truncated measures satisfy the box inequality and have finite diagrams.Applying the finite theorem to these restrictions produces partial matchings that can be limited over expanding subsets.

5. Examples

The examples apply the framework to partial interleavings and extended persistence. Truncation converts partial interleavings into ordinary ones, while weaker hypotheses suffice to define extended persistence diagrams.

  • Partial interleavings: Partial interleavings up to time t0 yield matchings in which points near the diagonal or truncation boundary need not be matched.All other points must be matched, and matched points have p-coordinates differing by at most δ.
  • Partial interleavings: Truncating U and V at T=t0+δ transforms diagram points above T and removes points born after T.The transformed diagram is obtained directly from the decorated diagram of the original module.
  • Partial interleavings: The truncated modules are δ-interleaved, so ordinary stability lifts to the stated partial matching between the original diagrams.The proof proceeds by diagram transformation, interleaving of truncations, and lifting the resulting matching.
  • Extended persistence: Extended persistence combines ordinary, relative, and extended features from sublevelset and superlevelset data.The three feature types are born and die respectively before, after, or across the central X.
  • Extended persistence: For a compact polyhedron with a continuous function, weaker rank-finiteness conditions suffice for the extended persistence diagram to be q-tame and defined away from the diagonal.The standard finite-critical-point assumption is therefore not required in this setting.
Loading 1207.3674v3…